A probability measure is a measure μ (on a σ-algebra A on a set A) such that μ(A)=1.
A measure on a σ-algebra A is a function μ:A→R with properties like “if A∩B=∅ then μ(A∪B)=μ(A)+μ(B)” etc.; the idea is that the elements of A are the subsets of A that are well-enough behaved to be “measurable” and then if X is such a subset then μ(X) says how big X is.
A σ-algebra on a set A is a set A of subsets of A that (1) includes all-of-A, (2) whenever it includes a set X also includes its complement A−X, and (3) whenever it includes all of countably many sets Xi also includes their union.
And now probability theory is the study of probability measures. (So the measure-theoretic definition of “probability” would be “anything satisfying the formal properties of a probability measure”, just as the mathematician’s definition of “vector” is “anything lying in a vector space”.)
“Bayesian” probability theory doesn’t disagree with any of that; it just says that one useful application for (mostly the more elementary bits of) the theory of probability measures is to reasoning under uncertainty, where it’s useful to quantify an agent’s beliefs as a probability measure. Here A is the set of ways the world could be; A is something like the set of sets of ways the world could be that can be described by propositions the agent understands, or the smallest σ-algebra containing all of those; μ, more commonly denoted P or P or P or something of the sort, gives you for any such set of ways the world could be a number quantifying how likely the agent thinks it is that the actual state of the world is in that set.
You can work with probability measures even if you think that it’s inappropriate to use them to quantify the beliefs of would-be rational agents. I guess that’s PP’s position?
What is the measure theoretic definition?
A probability measure is a measure μ (on a σ-algebra A on a set A) such that μ(A)=1.
A measure on a σ-algebra A is a function μ:A→R with properties like “if A∩B=∅ then μ(A∪B)=μ(A)+μ(B)” etc.; the idea is that the elements of A are the subsets of A that are well-enough behaved to be “measurable” and then if X is such a subset then μ(X) says how big X is.
A σ-algebra on a set A is a set A of subsets of A that (1) includes all-of-A, (2) whenever it includes a set X also includes its complement A−X, and (3) whenever it includes all of countably many sets Xi also includes their union.
And now probability theory is the study of probability measures. (So the measure-theoretic definition of “probability” would be “anything satisfying the formal properties of a probability measure”, just as the mathematician’s definition of “vector” is “anything lying in a vector space”.)
“Bayesian” probability theory doesn’t disagree with any of that; it just says that one useful application for (mostly the more elementary bits of) the theory of probability measures is to reasoning under uncertainty, where it’s useful to quantify an agent’s beliefs as a probability measure. Here A is the set of ways the world could be; A is something like the set of sets of ways the world could be that can be described by propositions the agent understands, or the smallest σ-algebra containing all of those; μ, more commonly denoted P or P or P or something of the sort, gives you for any such set of ways the world could be a number quantifying how likely the agent thinks it is that the actual state of the world is in that set.
You can work with probability measures even if you think that it’s inappropriate to use them to quantify the beliefs of would-be rational agents. I guess that’s PP’s position?
I see that gjm has replied with beautifully typeset text.